1. (20 pts.) Efficiency experts study the processes used to manufacture items in order to make them as efficient as possible. One of the steps used to manufacture a metal clamp involves the drilling of three holes. In a sample of 75 clamps, the mean time to complete this step was 42.4 seconds. Assume that the population standard deviation is 8 seconds.
(a) Construct a 98% confidence interval for the population mean. (b) Find the sample size needed so that a 99.9% confidence interval will have margin of error of 1.5.
2. (20 pts.) A news poll conducted in October 2009 surveyed a random sample of
1000 adults in the United States. Of these people, 710 said they would support federal legislation putting limits on the amounts that top executive are paid at companies that receive emergency government loans. One highly paid executive claims that the percentage of U.S. adults who support limits on the amounts that executives are paid is less than 72%. Use p-value method.
(a) (5 pts.) State the appropriate null and alternative hypothesis.
(b) (10 pts.) Find test statistics and P-value. (c) (5 pts.) State the conclusion.
3. (20 pts.) The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes.
(a) What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes?
(b) Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time?
4. (20 pts.) The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is 0.15.
(a) Show the sampling distribution of 𝑝̂ if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket.
(b) What is the probability that the sample proportion will be within ∓0.03 of the population proportion?
5. (20 pts.) A researcher conducts an IQ test every time she analyses a new data set. Over time, she conducts 200 tests.
(a) Suppose the null hypothesis is true in every case. What is the distribution of
the number of times she rejects the null hypothesis at the level 0.02 level? (b) Suppose she rejects the null hypothesis in eight of the tests. Is it plausible that the null hypothesis is correct in every case? Explain.